I saw the contest as well - but decided not to enter. However, my first thought was to use my caliper on the cases in the photograph - starting with the ones I was sure of. Not that your measurement would indicate correct size, but rather help establish size/length relative to the other cases.

I could then determined the approx. caliber of each by measuring the mouth. Once I had the caliber, then the length of the case, the shape and shoulder, plus other marks (double radius shoulder is telltale Weatherby) and rims (belted, rebated, etc) should narrow it down considerably.

The parent cartridge of the BRX is the BR, the BR is 1.075 base to shoulder, is the 6mm PPC the 221 is 1.070 base to shoulder, this removes the BR and PPC from the list of possibles as the one pictured is obviously longer base to shoulder. The BRX will naturally have a shorter neck than a BR because it is .10" longer to the shoulder than its parent. Fire form it and you will have a shorter neck for obvious reasons. The other possibility was a dasher but with a 40 degree neck that throws it out. And therefore it is a 6mm BRX, no other well known options.

By the way the picture I posted came from Ammoguide.com. They have a cartridge visual comparison tool. and most known cartridges in their database.